LATERAL MOVEMENT CONTROL DESIGN WITH ACCELERATION AND LANE ANGLE ALGORITHMS FOR FWS GROUND VEHICLES

B.-F. Wu and S.-M. Chang

References

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  11. [11] J.Y. Wang, The theory of ground vehicles (New York: Wiley,2001).278Appendix AThe system dynamics model of the vehicle motion isgiven by:˙β˙γ˙φ˙ϕ˙yL˙δC=a11 a12 0 0 0 b11a21 a22 0 0 0 b21a11 1 + a12 0 0 0 b110 1 0 0 0 00 0 U 0 0 0a61 a62 0 0 0 0βγφϕyLδC+b11b21b1100b61KLF rpath (A.1)where a61 = a11(1 − Ukmdc) + a21Lf ((1/U) − kmdc), a62 =a12(1 − Ukmdc) − Ukmdc + a22Lf ((1/U) − kmdc) and b61 =b11(1 − Ukmdc) + b21Lf ((1/U) − kmdc) + KP ref kmdc. Thevehicle motions are shown asβ(s)rpath(s)=NβD,γ(s)rpath(s)=NγD,φ(s)rpath(s)=Nφs · D,ϕ(s)rpath(s)=Nϕs · D,yL(s)rpath(s)=NyLs2 · DandδC(s)rpath(s)=NδCDwhereNβ = KLF Cf (−mU2Lf + CrLrL + mLf LrUs)(Cf L + mULrs + kmdcKP ref mLrU − Cf LkmdcU)Nγ = UKLF Cf (mLf Us + CrL)(Cf L + mULrs− Cf kmdcUL + kmdcKP ref mLrU)Nφ = KLF Cf (mULf Lrs2+ CrLrLs + CrUL)(mLrUs+ Cf L − UCf kmdcL + kmdcKP ref mLrU)Nϕ = Nγ, NyL= UNφNδC= LrKLF Cf mU(−mU2Cf Lf kmdcL+ Lrm2U2Lf kmdcKP ref + mULf Cf L)s2+ (−Cf CrkmdcUL2+ Cf CrL2− Cf Lf mU2+ LrLmUCrkmdcKP ref )s+ (CrLr − Cf Lf )mU2kmdcKP ref+ L2Cf CrkmdcKP ref − CrLU2Cf kmdcandD = m3U3L2rLf s3+ m2U2LrL(Cf Lf + CrLr)s2+ [mUCf L2(CrLr + Cf Lf ) − mC2f U2Lf kmdcL2+ CrL2rm2U3]s + [C2f CrL3− kmdcUC2f CrL3+ mU2Cf L(CrLr − Cf Lf ) + mC2f kmdcU3Lf L]The state variable ˙δC, derived from (24), is expressedas:˙δC =afU+ kmdc(KP ref δS − af ) − γ = kmdcKP ref δS+1U− kmdcaf − γ = kmdcKP ref δS+ (1 − Ukmdc)( ˙β + γ) +1U− kmdcLf ˙γ − γ= kmdcKP ref δS + (1 − Ukmdc) ˙β + (1 − Ukmdc)γ+1U− kmdcLf ˙γ − γ= kmdcKP ref δS + (1 − Ukmdc)(a11β + a12γ + b11δS)+ (1 − Ukmdc)γ +1U− kmdcLf (a21β + a22γ+ b21δS) − γ=a11(1 − Ukmdc) + Lf a211U− kmdcβ+a12(1 − Ukmdc) − Ukmdc + Lf a221U− kmdcγ+b11(1 − Ukmdc) + Lf b211U− kmdc+kmdcKP refδSAppendix BThe vehicle model is shown in (B.1) as:˙β˙γ =a11 a12a21 a22βγ +b11b21δC (B.1)Furthermore, when (B.1) is transferred to (B.2) with thefirst differentiation of the sideslip angle at the front axle( ˙βf = ˙β + (Lf /U)˙γ), the state variables ˙βf and ˙γ can beobtained:˙βf −LfU ˙γ˙γ =a11 a12a21 a22βf −LfU γγ +b11b21δC(B.2)Rearranging (B.2) to yield:˙βf −LfU ˙γ˙γ =a11 a12 − a11LfUa21 a22 − a21LfUβfγ +b11b21δC(B.3)and multiplying by the factor Lf /U into ˙γ, it obtains˙βf −LfU ˙γLfU ˙γ =a11 a12 − a11LfUa21LfU (a22 − a21LfU )LfUβfγ+b11b21LfUδC (B.4)279To eliminate (Lf /U)˙γ, two state equations in (B.4) areincorporated as:˙βfLfU ˙γ =a11 + a21LfU a12 − a11LfU + (a22 − a21LfU )LfUa21LfU (a22 − a21LfU )LfU×βfγ +b11 + b21LfUb21LfUδC (B.5)Finally, the second state ˙γ in above expression is dividedby the factor (Lf /U) resulting in˙βf˙γ =a11 + a21LfU a12 − a11LfU + a22LfU − a21LfU2a21 a22 − a21LfU×βfγ +b11 + b21LfUb21δC (B.6)The relationship, I := mLf Lr, is included in (B.6) andthen its expression can be simplified as:˙βf˙γ =−2Cf LmULr−1−2 (Lf Cf − LrCr)mLf Lr−2CrLmLf Uβfγ+2Cf LmLrU2CfmLrδC (B.7)Realigning (B.7) and considering the kinematics rela-tionship, βf = βr + (L/U)γ, we obtain˙βf˙γ =2Cf LmULr02CfmLr−2CrmLfδC − βf−βr−10γ (B.8)

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